Numerical Solution of Monge-Kantorovich Equations via a dynamic ...

NUMERICAL SOLUTION OF MONGE-KANTOROVICH EQUATIONS VIA A DYNAMIC FORMULATION

arXiv:1709.06765v1 [math.NA] 20 Sep 2017

ENRICO FACCA, SARA DANERI, FRANCO CARDIN AND MARIO PUTTI

Abstract. We propose a biologically inspired dynamic model for the numerical solution of the L1 -PDE based optimal transportation model. Starting from the conjecture that the continuous model describing the dynamics of slime mold presented in [12] is time-asymptotically equivalent to the MongeKantorovich equations governing L1 optimal transport, we propose a simple yet effective method for its numerical solution. The transport dynamics is described by a nonlinear system of equations coupling an elliptic partial differential equation for the Kantorovich potential with an ordinary differential equation for the transport density µ. The novel contributions in this paper are twofold. First, we introduce a new Lyapunov candidate functional that better adheres to the dynamics of our proposed model. We calculate its Lie-derivative and show it is always negative along µ-trajectories and attains zero at the equilibrium. The conditions arising by setting to zero this Lie-derivative more strongly resemble the M-K equations. Second, we describe and test different numerical approaches for the solution of our problem. The ordinary differential equation for the transport density is projected into a piecewise constant or linear finite dimensional space defined on a triangulation of the domain. The elliptic equation is discretized using a linear Galerkin finite element method defined on uniformly refined triangles. The ensuing nonlinear differential-algebraic equation is discretized by means of a first order Euler method (forward or backward) and a simple Picard iteration is used to resolve the nonlinearity. The procedure is iterated in time until relative differences on the spatial norm of the transport density are smaller than a predefined tolerance. The use of two discretization levels is dictated by the need to avoid oscillations on the potential gradients that prevent convergence of the scheme. We study the experimental convergence rate of the proposed solution approaches and discuss limitations and advantages of these formulations. An extensive set of test cases, including problems that admit an explicit solution to the Monge-Kantorovich equations so that error norms can be accurately evaluated, are appropriately designed to verify and test the expected numerical properties of the solution methods. The results show optimal convergence toward the asymptotic equilibrium point is achieved for sufficiently regular forcing function. All the obtained numerical solutions support the conjecture that indeed the dynamic model possess a time-asymptotic equilibrium point that coincides with the solution of the Monge-Kantorovich equations. Also the existence and coherence of the new Lyapunov-candidate function is confirmed.

Monge-Kantorovich equations, Optimal Transport, Numerical Solution [2000] 35M20 65M60 65M12

1. Introduction We are interested in finding the numerical solution to the following nonlinear differential problem. Given a domain Ω ⊂ Rn , a function f : Ω → R such that R f dx = 0, and a function k : Ω → R+ , find the pair (µ, u) : [0, +∞[×Ω 7→ R+ × R Ω 1

2


that satisfies: (1a) (1b) (1c)

− ∇· (µ(t, x) ∇ u(t, x)) = f (x)

µ0 (t, x) = µ(t, x) (| ∇ u(t, x)| − 1)

µ(0, x) = µ0 (x) > 0

complemented by homogeneous Neumann boundary conditions for eq. (1a). Here, µ0 indicates partial differentiation with respect to time and ∇ = ∇x indicates the spatial gradient operator. This model, proposed initially in Facca et al. [12], is a generalization to the continuous setting of the discrete model proposed in Tero et al. [17] for the simulation of the dynamics of Physarum Polycephalum, a slime mold with exceptional optimization abilities [14]. More recently, this latter model was analyzed in Bonifaci et al. [6], who proved its equivalence to an optimal transportation problem on a graph. In analogy to the discrete setting, [12] conjecture that, at infinite times, model eq. (1) is equivalent to the PDE-based formulation of the Monge-Kantorovich (MK) optimal transportation problem as given in [11]: find (µ∗ , u∗ ) ∈ L∞ (Ω) × Lip1 (Ω), with Lip1 (Ω) the space of Lipschitz continuous functions with unit constant, such that: (2a) (2b) (2c)

− ∇·(µ∗ (x) ∇ u∗ (x)) = f (x)

| ∇ u∗ (x)| ≤ 1 ∗

| ∇ u (x)| = 1

∀x∈Ω

where µ∗ (x) > 0

Isolating eq. (2a), the function µ∗ , called the transport density, under proper assumptions on the forcing function and on the domain, is uniquely defined by f , and in the sequel we will often use the notation µ∗ (f ) when the x dependence is not needed explicitly. The function u∗ is called the transport or Kantorovich potential Villani [18]. We will term this problem as the L1 -MK equations (or simply MK equations) problem to distinguish it from the L2 -MK problem characterized by a quadratic distance cost function. The analysis of system eq. (1) was performed in [12] by transforming eq. (1) into an ODE in Banach spaces by defining the operator U(µ) as the weak solution of eq. (1a) given µ and the corresponding flux operator Q(µ) = µ |∇ U(µ)|. The problem can then be rewritten as: (3)

µ0 (t) = Q(µ(t)) − µ(t) µ(0) = µ0

Using this setting, the authors were able to prove local in time existence and unique¯ In addition ness of the solution under the assumption of f ∈ L∞ (Ω) and µ0 ∈ C δ (Ω). to local existence, the authors propose a Lyapunov-candidate function, i.e., a scalar functional that decreases along the µ-trajectories. This functional takes the form: Z Z L(µ) = µ dx · µ| ∇ U(µ)|2 dx Ω

Ω

and it can be proved, assuming global existence and uniqueness, that its Lie derivative is strictly negative in time. Although existence of the solution can be shown only locally, the conjecture that the two problems eq. (1) and eq. (2) are equivalent at infinite times is supported by strong indications, including numerical evidence obtained by solving the test cases proposed in Barrett and Prigozhin [1]. The main difficulty in proving existence and uniqueness of the solution to eq. (1) at large times, which would enable the proof of the conjecture, seems to be related to the fact that the coercivity constant of the elliptic equation eq. (1a) varies in time, thus allowing only locally-Lipschitz estimates for the Kantorovich potential and the flux operators. However, the lack of

NUMERICAL SOLUTION OF MONGE-KANTOROVICH EQUATIONS

3

coercivity is localized where µ → 0, and the simple trick of setting a lower bound on the numerically evaluated transport density yields efficient and robust simulations at all times. This suggests that the dynamic optimal transport problem can be an effective strategy for the numerical solution of the L1 -MK equations. Unlike the L2 case, for which the noteworthy computational fluid mechanic formulation of [2, 3] allows for efficient numerical solution, discretization of the L1 formulation treated in this study is much more complicated, being based on nonlinear minimization algorithms [1] or on the solution of the highly nonlinear MongeAmpere equation on the product space with regularization [8–10]. On the other hand, discretization of the dynamic model is straight forward and efficient, and is proposed here as an alternative formulation amenable of robust and efficient numerical solution without introduction of additional regularizing parameters. Note that we can think of time as a regularization term but we prefer to think of it as a dynamic version of the overall mass transport phenomenon. In this paper, we first introduce the following new Lyapunov-candidate functional: Z Z 1 1 µ| ∇ U(µ)|2 dx + µ dx (4) S(µ) := 2 Ω 2 Ω We show that this functional is strictly decreasing along µ-trajectories and its Lie derivative annihilates at equilibrium, highlighting the MK gradient constraint within the support of µ∗ . However, the requested uniform bound on | ∇ u∗ | in Ω does not yet surface from this calculations. This new Lyapunov-candidate functional seems to adhere more closely to the dynamics of our ODE eq. (3) and provides √ a uniform bound on L since L ≤ S for all µ. After this brief but important interlude on the theoretical analysis of our dynamic model, we focus on an extensive experimental analysis with the twofold ambition of i) supplying additional support of the equivalence conjecture introduced in Facca et al. [12], and ii) to corroborate the thesis that this dynamic MK model provides an ideal setting for the numerical solution of the L1 -MK equations. To this aim, we derive several numerical approaches for the solution of eq. (1). All the considered methods couple together simple and cost-effective low order (P1 or triangular P0 ) Galerkin finite element spaces with Euler (forward or backward) scheme for the time-discretization of the ensuing Differential Algebraic (DAE) system of equations [16]. Successive (Picard) iterations are used when necessary to resolve the nonlinearities. The expected convergence of the different approaches are tested at large simulation times against the closed form solution proposed by [7] for sufficiently regular forcing functions. We also verify the convergence toward steady-state for increasingly refined grids and the behavior of the proposed Lyapunov-candidate functions. Finally, we extend the comparison already presented in Facca et al. [12] of our model results against those reported in Barrett and Prigozhin [1]. The test considers a sequence of spatially refined grids where we look at monotonicity of the solution and convergence of the Lyapunov-candidate functions toward a common value. The numerical results show that the use of one single grid may promote the emergence of oscillations in the numerical gradient field. These oscillations are amplified by the companion ODE solver, eventually preventing the long-time convergence of the schemes. A monotone solution is obtained by discretizing the transport potential on a triangulation that is uniformly refined (Th/2 ) with respect to the triangulation Th , where the gradient field and the transport density are defined, similarly to what happens with the inf-sup stable mixed finite elements methods for the solution of Stokes equation [5]. In the case of a mesh aligned with the support

4


of the transport density, optimal grid convergence for smooth (C 1 ) forcing functions is obtained using either P0 (Th ) or P1 (Th ) for transport density and gradient, i.e., using P1 (Th/2 ) to discretize the transport potential and either the P1 (Th ) or P0 (Th ) for the transport density. For piecewise continuous forcing function, the loss of convergence seems to affect more the P1 -P1 strategy than P1 − P0 , which seems to be more robust. When the grid is not aligned with the support of the transport density, errors due to geometrical approximations dominate and optimal convergence is again lost even for smooth forcing functions. Grid refinement is recommended to solve this problem. However, we work on fixed grids, because we are interested in ascertaining the convergence properties of the base methods. In the case of spatially heterogeneous domain, the comparison against the published numerical results is qualitatively coherent, but no quantitative result is obviously possible. However, convergence of the Lyapunov-candidate functions toward the equilibrium point is verified. 2. The Lyapunov-candidate Functional S We show here that, under some existence assumptions, the new functional S defined in eq. (4) is strictly decreasing in time. In fact, its Lie derivative along the µ(t)-trajectory can be calculated using the same procedure described in [12] leading to: Z 1 d 2 S(µ(t)) = − µ(t) (| ∇ U(µ(t))| − 1) (| ∇ U(µ(t))| + 1) dx (5) dt 2 Ω This quantity is strictly negative ∀t ≥ 0 and is equal to zero only if | ∇ U(µ(t))| = 1 within the support of µ(t). Moreover, a simple application of Young inequality shows that p L(µ) ≤ S(µ) for all µ > 0

Remark 1. Note that if µ = µ∗ , then | ∇ u∗ | = 1 and thus the Lie derivative of S is zero. This fact strengthens the conjecture that the infinite time solution of our dynamical model is equivalent to the MK-equations, at least within the support of µ∗ . However, the uniform bound on | ∇ u∗ | does not seem to be explicitly imposed by our model. Intuitively, this could be a consequence of the fact that the MK-potential u∗ is not uniquely defined outside the support of µ∗ [11]. We would like to point out that the Lie derivative of the functional L reported in [12] does not share these properties. 3. Numerical discretization We start this section by stressing the fact that our aim is to show the effectiveness of simple discretization methods for the solution of eq. (1). Obvious improvements in both computational efficiency and accuracy can be obtained by using more advanced approaches, such as, e.g., higher order approximations, Newton method, automatic mesh refinement, etc. However, our starting point is to show that even the simple methods presented here form an efficient and robust framework for the solution of the L1 -MK equations using the proposed dynamic setting. 3.1. Projection spaces. Our numerical approach at the solution of eq. (1) is based on the method of lines. Spatial discretization is achieved by projecting the weak formulation of the system of equations onto a pair of finite dimensional spaces (Vh , Wh ). We denote with Th (Ω) a regular triangulation of the (assumed polygonal) domain Ω, characterized by n nodes and m triangles, where h indicates the characteristic length of the elements. We also denote with P0 (Th (Ω)) = span{ψ1 (x), . . . , ψM (x)} the space of element-wise constant functions on Th (Ω), i.e., ψi (x) is the characteristic function of triangle Ti , and with P1 (Th (Ω)) =


5

span{ϕ1 (x), . . . , ϕN (x)} the space of element-wise linear Lagrangian basis functions defined on Th (Ω). We consider two different choices of the space Vh used in the projection of the elliptic equation eq. (1a), namely Vh = P1,h = P1 (Th (Ω)) and Vh = P1,h/2 = P1 (Th/2 (Ω)). Here Th/2 (Ω) is the triangulation generated by conformally refining each triangle Tk ∈ Th (Ω). Again we consider different choices of spaces also for the projection of the dynamic equation eq. (1b) by using alternatively Wh = P1,h and Wh = P0,h = P0 (Th (Ω)), when the projection is done on the same mesh used for the elliptic equations, or Wh = P1,h/2 Wh = P0,h = P0 (Th (Ω)), when we use the sub-grid. Following this approach and separating the temporal and spatial variables, the discrete potential uh (t, x) and diffusion coefficient µh (t, x) are written as: uh (t, x) =

N X i=1

ui (t)ϕi (x) ϕi ∈ Vh

µh (t, x) =

M X k=1

µk (t)ψk (x) ψk ∈ Wh

where N and M are the dimensions of Vh and Wh , respectively. The finite element discretization yields the following problem: for t ≥ 0 find (uh (t, ·), µh (t, ·)) ∈ Vh × Wh such that Z Z (6a) aµh (uh ,ϕj ) = µh ∇ uh · ∇ ϕj dx = (f, ϕj ) = f ϕj dx j = 1, . . . , N Ω Ω Z Z (6b) µ0h ψl dx = (|µh ∇ uh | − µh )ψl dx l = 1, . . . , M Ω Ω Z Z (6c) µh (0, ·)ψj dx = µ0 ψl dx l = 1, . . . , M Ω

Ω

R where we add to eq. (6a) the zero-mean constraint Ω uh dx = 0 to enforce wellposedness. In matrix form, indicating with u(t) = {ui (t)}, i = 1, . . . , N , and µ(t) = {µk (t)}, k = 1, . . . , M , the vectors that describe the time evolution of the projected system, we can write the following index-1 nonlinear system of differential algebraic equations (DAE): (7a) (7b)

a · u(t) = 0

A[µ(t)] u(t) = b, 0

M µ (t) = B(u(t)) µ(t),

M µ(0) = µ0

where the N × N stiffness matrix A[µ(t)] is given by: Z M X Aij [µ(t)] = µk (t) ψk ∇ ϕi · ∇ ϕj dx, Ω

k=1

R b is the N -dimensional source vector whose components R are bi = Ω f ϕi dx, a is the N -dimensional vector with elements given by ai = Ω ϕi dx, the M × M mass matrix M is expressed by: Z Mk,l =

ψk ψl dx, Ω

the M × M matrix B has the same structure of M and is defined as ! Z N X Bk,l [u(t)] = | ui (t) ∇ ϕi | − 1 ψk ψl dx Ω

i=1

and the M -dimensional vector µ0 contains the projected initial condition µ0,l = R µ ψl dx. When we consider Wh = P0,h , matrices M and B are diagonal 0 Ω and eq. (7) simplifies in: (8a) (8b)

A[µ(t)]u(t) = b, 0

µ (t) = D[u(t)]µ(t),

a · u(t) = 0

µ(0) = µ ˜0

6


where D is the M × M diagonal matrix given by: ! Z N X 1 Dk,k [u(t)] = | ui (t) ∇ ϕi | − 1 dx |Tk | Tk i=1 where |Tk | the size of the element ˜ 0 is the M -dimensional vector with R Tk , and µ components given by µ ˜0k = |T1k | Tk µ0 dx. 3.2. Time discretization. In order to solve the DAE eq. (7) or eq. (8) we operate a discretization in time using either a forward or a backward Euler scheme. Denoting with ∆tk the time-step size so that tk+1 = tk + ∆tk and (uk , µk ) = (u(tk ), µ(tk )), PN the approximate solutions at time tk can be written as ukh (x) = i uki ϕi (x) and P M µkh (x) = l=1 µkl ψl (x). 3.2.1. Case µh (t, ·) ∈ P1,h . In this case, the forward Euler scheme yields: A[µk ] uk = b,

a · uk = 0

µk+1 = (I + ∆tk M −1 B[uk ])µk ,

µ0 = M −1 µ0

When backward Euler is employed, the time-stepping scheme becomes: A[µk+1 ]uk+1 = b,

a · uk+1 = 0

µ0 = M −1 µ0

M µk+1 = M µk + ∆tk B[uk+1 ]µk+1 ,

and the nonlinearity is resolved by means of the following successive iteration (Picard) scheme: µ0,k+1 = µk for m = 0, 1, 2, . . . A[µm,k+1 ]um,k+1 = b, (9)

a · um,k+1 = 0

µm+1,k+1 = (M − ∆tk B[um,k+1 ])−1

M µk

which can be repeated until the relative difference ρ(µm+1,k+1 , µm,k+1 ) ≤ τNL , h h where kµhm+1,k+1 − µm,k+1 kL2 (Ω) m,k+1 h ρ(µm+1,k+1 , µ ) = h h m,k+1 kµh kL2 (Ω) or the number of Picard iterations m reaches a prefixed maximum mMAX . 3.2.2. Case µh (t, ·) ∈ P0,h . In this case the forward Euler scheme reads: A[µk ] uk = b,

a · uk = 0

µk+1 = (I + ∆tk D[uk ])µk ,

µ0 = µ ˜0

while the backward Euler discretization yields the non-linear system of equations: A[µk+1 ] uk+1 = b, µk+1 = µk + ∆tk

a · uk+1 = 0 D[uk+1 ]µk+1 ,

µ0 = µ ˜0

solved again by Picard iteration: µ0,k+1 = µk A[µm,k+1 ]um,k+1 = b, (10)

a · um,k+1 = 0

µm+1,k+1 = (I − ∆tk D[um,k+1 ])−1 µk


7

Figure 1. Domain Ω and supports Q+ , Qc , and Q− for Test Case 1. Unrefined initial meshes. From the left to the right: Mesh 1 (constrained Delaunay, 438 nodes and 810 elements), and Mesh 2 (constrained Delaunay, 297 nodes and 528 elements). The edges of Mesh 1 are aligned with the supports of f and µ∗ (f ). Mesh 2 is aligned only with the supports of f . repeated until ρ(µm+1,k+1 , µm,k+1 ) ≤ τNL or m > mMAX . Note again that the h h matrix (I − ∆tk D) in eq. (10) is trivially invertible being diagonal. 3.3. Solution of the linear system. At each Picard iteration we need to solve a large sparse symmetric linear system that is positive semi-definite because homogeneous Neumann boundary conditions are used. We solve the system by a preconditioned conjugate gradient (PCG) method and employ the approach suggested by Bochev and Lehoucq [4] to construct the Krylov subspace orthogonal to the null space of the system matrix. Convergence of the PCG iteration is considered achieved when the Euclidean norm of the residual relative to the initial residual norm is smaller than the tolerance τCG . We start the iteration from ukh , i.e., solution at the previous time step and use an incomplete Choleski factorization with no fill-in as preconditioner. Since the system dynamics drives the transport density µh to zero in large portions of the domain Ω, we set a lower limit to it and impose that µh ≥ 10−10 everywhere. This is sufficient to guarantee the ellipticity of the FEM bilinear form and allows the system condition number to remain bounded so that PCG converges within a limited number of iterations. 4. Numerical experiments 4.1. Test Case 1: comparison with closed-form solutions. In this first set of tests we experiment the convergence of the different numerical schemes toward the optimal transport density by comparing the numerical solution with the closed-form solution of the Monge-Kantorovich equations discussed in Buttazzo and Stepanov [7]. To this aim, we consider a square domain in R2 , Ω = [0, 1] × [0, 1], and a zero-mean forcing function f supported in two rectangles Q+ and Q− contained in Ω, where f assumes opposite signs (fig. 1). The different supports are given by: 1 3 1 3 + , × , Q = (x, y) ∈ Ω : (x, y) ∈ 8 8 4 4 1 3 5 3 c × , Q = (x, y) ∈ Ω : (x, y) ∈ , 8 8 4 4 5 7 1 3 − Q = (x, y) ∈ Ω : (x, y) ∈ , × , 8 8 4 4

8


We are optimally transporting f + = f (Q+ ) into f − = f (Q− ) and look for the density µ(t, x) and the potential u(t, x) that satisfy eq. (1) as t → ∞ and approximate µ∗ solution of eq. (2). We consider that a time-equilibrium condition has been achieved when the relative variation in µh (var(µh )) is smaller than a tolerance, tstep i.e., var(µh ) := ρ(µk+1 )/∆tk < τT . We indicate with t∗ the time when time h , µh equilibrium is numerically reached and with µ∗h the corresponding µkh . To test our numerical schemes we set up two different problems that differentiate by the specific choice of f . In particular, the first test case considers a continuous forcing function f1 with opposite sign in Q+ and Q− , while the second case considers a piecewise constant function f2 :  1 1   sin 2π y − in Q+ 2 sin 4π x −   8 4   5 1 f1 (x, y) = − 2 sin 4π x − sin 2π y − in Q−    8 4    0 elsewhere and  +   2 in Q f2 (x, y) = − 2 in Q−   0 elsewhere Correspondingly, the optimal transport density µ∗ (f ) is given by [7]:  1 1 1   1 − cos 4π x − sin 2π y −    2π 8 4       1 sin 2π y − 1 in Qc 4 µ∗ (f1 )(x, y) = π    1 7 1   1 − cos 4π −x sin 2π y −   2π 8 4     0 elsewhere

in Q+

in Q−

and  1   2 x− in Q+   8       1 in Qc ∗ µ (f2 )(x, y) = 2   7    2 −x in Q−   8    0 elsewhere Note that the support of µ∗ (f ) is given by Qµ = Q+ ∪ Qc ∪ Q− . With this explicit solution, we can verify the experimental convergence rates in space at infinite times for the different proposed schemes. We use two different initial triangulation settings (Mesh 1 and Mesh 2, fig. 1) that are uniformly refined four times to assess FEM convergence. Both meshes are constrained P R to be aligned with the exact supports of f + and f − , so that the condition i Ω f (x)ϕi dx = 0 can be imposed exactly, and have approximately the same number of nodes and elements. Mesh 1 (fig. 1, left) is a constrained Delaunay triangulations with edges aligned with the boundary of Qµ . Mesh 2 is also a constrained Delaunay triangulation but is not aligned with Qµ in the area between Q+ and Q− . Hence, in the latter case, we expect convergence to be influenced also by the geometric convergence of the mesh element boundaries toward the support Qµ of µ∗ . Sensitivity to initial conditions


9

(i)

is tested by employing the following different initial data µ0 : (1)

µ0 (x, y) = 1; (11)

(2) µ0 (x, y) = 0.1 + 4 (x − 0.5)2 + (y − 0.5)2 ; (3)

µ0 (x, y) = 3 + 2 sin(8πx) sin(8πy). Note that in these tests we are not interested in computational speed, but only on the numerical behavior of the schemes. Thus we do not limit the minimum time step size and the maximum number of iterations (in both time-stepping and the PCG algorithm used to solve the linear system of algebraic equations), and use a very tight tolerance to determine when time equilibrium is reached. We employ very tight tolerances, i.e., τNL = 10−11 and τT = 5×10−9 , and a PCG exit tolerance on the relative residual τCG = 10−13 . We would like to remark that much more efficient simulations are obtained with much higher tolerances (≈ 10−3 ) maintaining meaningful quantitative results. In the simulations presented here we set m = 1 (only one Picard iteration is performed in each time step) and vary ∆tk by setting ∆tk+1 = min(1.05 × ∆tk , ∆tmax ), where ∆tmax = 0.5 was found experimentally to ensure the stability of the forward Euler scheme, or equivalently, the convergence of the Picard iteration. Convergence as h → 0 is explored by looking at the time behavior of the L2 (Ω) relative error defined as: err(µh (t), f ) :=

kµh (t) − µ∗ (f )kL2 (Ω) kµ∗ (f )kL2 (Ω)

Convergence toward steady-state equilibrium. We analyze the experimental convergence toward an equilibrium point (µ∗h , u∗ ) by looking at the time evolution of var(µh (t)) and err(µh (t)). Figure 2 reports the log-log scale plots of these two quantities calculated in the case of continuous forcing function. The different curves in each sub-plot display the behavior obtained on successive uniform refinements of the two different initial meshes of fig. 1, while the columns identify the different combinations of spatial discretizations. Only results of the Explicit Euler time-stepping scheme are shown, the results of the Implicit Euler method being identical. The first set of plots (first two rows) are relative to the Qµ -aligned mesh set, while the lower set reports the results for the Qf -aligned meshes. The µh variation, var(µh (t)), displays a monotone behavior for all schemes, with an expected geometric convergence rate toward steady-state, as evidenced by the slope of the rectilinear portions of the curves which coincides for all mesh-levels and for both mesh types. At increasing refinement levels the plots deviate as a consequence of the higher accuracy of the spatial discretization. This is testified also by the fact that the convergence curves for all schemes initially coincide, starting to diverge approximately when the corresponding spatial accuracy limit is attained. Accuracy saturation in the error plots (err(µh (t)) vs. t) occurs at the same time at which var(µh (t)) start diverging. More uncertain profiles are obtained when spatial discretization is performed on the same mesh for the pair (µh , uh ) for both P1 − P0 and P1 − P1 discretization spaces. The reason for the loss of regularity is to be attributed to oscillations in the cell gradients that cause amplified oscillations in the corresponding transport density. Spatial average of the gradient magnitudes, leading to the Th −Th/2 , shows a much smoother behavior with a faster convergence towards equilibrium. We postpone a more detailed discussion of this phenomenon to section 4.2, where a more challenging test case is approached. Looking at the bottom half of fig. 2, we see the effect of using meshes that are not aligned with the support of the optimal transport density. Because of the discontinuity in µh occurring across the boundary of Qµ , convergence is limited by the geometric convergence of the triangular shapes towards this boundary, and

10


Qµ -aligned mesh (Mesh 1) var(µh (t))

P1,h/2 − P0,h

P1,h − P0,h h h/2 h/4 h/8

−2

10

P1,h/2 − P1,h

P1,h − P1,h h h/2 h/4 h/8

h h/2 h/4 h/8

h h/2 h/4 h/8

100 10−2

−4

10−4

−6

10−6

10 10

−8

10 err(µh (t)) 100 10−1 10−2 10−3 10−4

h h/2 h/4 h/8

10−1 101

103

105

h h/2 h/4 h/8

10−1 101

103

105

h h/2 h/4 h/8

10−1 101

103

105

h h/2 h/4 h/8

10−1 101

103

105 t

10−8 101 100 10−1 10−2 10−3 10−4

Qf -aligned mesh (Mesh 2) var(µh (t))

P1,h/2 − P0,h

P1,h − P0,h h h/2 h/4 h/8

10−2

P1,h/2 − P1,h

P1,h − P1,h h h/2 h/4 h/8

h h/2 h/4 h/8

h h/2 h/4 h/8

100 10−2

10−4

10−4

10−6

10−6

10−8 err(µh (t)) 100 10−1 10−2 10−3 10−4

h h/2 h/4 h/8

10−1 101

103

105

h h/2 h/4 h/8

10−1 101

103

105

h h/2 h/4 h/8

10−1 101

103

105

h h/2 h/4 h/8

10−1 101

103

105 t

10−8 101 100 10−1 10−2 10−3 10−4

Figure 2. Convergence toward equilibrium in the case of continuous forcing (f = f1 ). The log-log plots of var(µh (t, ·)) and err(µh (t, ·)) vs. time are reported for Mesh 1 (Qµ -aligned, top block) and for Mesh 2 (Qf -aligned, bottom block). The columns refer from left to right to the results obtained with P1,h − P0,h , P1,h/2 − P0,h , P1,h − P1,h , P1,h/2 − P1,h , respectively. the global attainable accuracy is bounded by this error. We observe a consistent behavior of the error for both mesh-types at different h levels. The accuracy levels at which the error saturates decrease consistently with the expected order of spatial convergence of the different schemes, when the geometric error is negligible. This is clearly observable by looking at the plots of var(µh (t)) for the P1,h/2 − P0,h , and the P1,h/2 − P1,h , cases, where the optimal second order convergence of the latter approach is observable from the fact that difference in the attained accuracy levels are doubled with respect to the first order approach. In the case of the Qf -aligned mesh, the higher order approach displays more accurate results with respect to the first order method, but the geometric error prevents the realization of optimal convergence rates. Convergence of the spatial discretization. We would like to recall that continuity of the transport density µ∗ (f ) when the forcing term f is continuous was proved in R2 in [13] under some assumptions over f . However, except for partial regularity results along transport rays [7], the general case seems to be open question. In our test cases, for both f1 and f2 forcings, strong variations in µh are present in a direction orthogonal to the boundary of Qµ in the central portion of the domain (outside Qf ). Because of these variations, which in the discontinuous forcing case are actual µ∗ -discontinuities, we expect a loss of convergence in the FE solution. We note, however, that convergence towards steady-state is not affected by spatial errors, as shown in the previous discussion.


h

h/2

h/4

h/8

h

h/2

h/4

11

h/8

err(µ∗h )

err(µ∗h )

10−1

10−1

10−2

10−2

10−3

P1,h − P0,h - 0.86 P1,h − P1,h - 1.89 P1,h/2 − P0,h - 1.00 P1,h/2 − P1,h - 2.11

10−4 err(µ∗h )

10−3


10−4 err(µ∗h )

10−1

10−1

10−2

10−2

10−3

10−4


h

h/2

10−3


h/4

h/8

h

h/2

h/4

h/8

10−4

Figure 3. Behavior of err(µ∗h ) vs. h for the different discretization methods. The results for the continuous forcing function f1 are shown in the left column, while the right column reports the results for f2 . The top row is relative to the Mesh-1 sequence (aligned with Qµ ), while the bottom row corresponds to the Mesh2 sequence (aligned only with Qf ). For visual reference, the first order convergence line is also plotted with a thick solid trait. The average experimental convergence rates are reported in the legends of each plot next to the discretization method.

The experimental convergence profiles for the different methods are reported in fig. 3. The column on the left groups the results relative to the more regular case raised by the continuous forcing function f1 (left). The right column reports the results obtained in the case with the piecewise constant forcing f2 . The top and bottom rows identify the mesh sequences aligned with the boundary of Qµ or with the boundary of Qf , respectively. From the two plots on the left, we can argue that all methods attain optimal convergence when the Mesh-1 sequence is used. The Th − Th/2 combination is characterized by a smoother behavior. The use of the Mesh-2 sequence, which we recall is aligned only with the boundaries of Qf and not those of Qµ , triggers the emergence of geometrical errors that cause a sizeable reduction on the convergence rates of both P1 − P1 and P1 − P0 schemes. As expected, the results for the discontinuous forcing function (right column) are characterized by an important loss of convergence rate for all schemes, except the P1,h/2 − P0,h in combination with the Qµ -aligned meshes. The loss of regularity in the solution due the lower regularity of f , together with the eventual geometric error in the Mesh-2 sequence, reduces the convergence rates of the schemes. The use of a Th − Th/2 combination seems to alleviate somehow the rate loss. This is confirmed by the spatial distribution of the error µ∗h − µ∗ (f2 ) shown in fig. 4. In this figure we report the results obtained with the P1,h − P0,h (upper left panel) and the P1,h − P1,h (upper right panel) for Mesh 1, and the P1,h/2 − P0,h (lower left panel) the P1,h/2 − P1,h (lower right panel) for Mesh 2. The plots suggest that

12


Figure 4. Spatial distribution of the error µ∗h − µ∗ (f2 ) at steady state for the piecewise constant forcing function f2 . The upper row reports the results on the finest level of Mesh-1 obtained with the P1,h − P0,h , (left) and P1,h − P1,h , (right). The lower row shows the results on the finest level of Mesh-2 from the P1,h/2 − P0,h , (left) and P1,h/2 − P1,h , (right) approaches. the P1 − P1 approach localizes the error on the north and south boundaries of Qµ , where the jump in µ∗ from 0 to 0.5 localizes. The P1 − P0 approach, on the other hand, displays an additional small but non negligible error on the support of the forcing function Qf . The zooms on the pictures show clear oscillations for the one-mesh methods (upper row) in both directions orthogonal and parallel to the µh -discontinuity. On the contrary, the methods based on two-meshes (lower row) exhibit a monotone error behavior along the boundary of Qµ , but the mis-alignment of the triangle edges causes an increased error as compared to the Mesh-1 results. The error slightly oscillates in the direction normal to the µh -jump due to the gradient reconstruction. It is evident that the smoothing due to the averaging of the gradient magnitude on the larger triangles helps in reducing overall oscillations. This will become more evident when we will discuss these oscillations in connection with a more challenging test case in section 4.2. Computational cost. We would like to remark here that our numerical approach is not optimized for computational cost. In fact, we are mainly interested in exploring the accuracy and feasibility of our solution approach. Nonetheless, it is interesting to include a discussion on computational cost in the worst-case scenarios described here. A number of cost-saving strategies could be envisaged, including using coarsemesh solutions to extrapolate initial guesses, re-use of stiffness and preconditioning matrices, developing a Newton-Raphson strategy to improve stability and employ larger time-step sizes, etc. On the other hand, here we are interested in showing


N it.

N it. P1 (h) − P0 (h)

800

P1 (h/2) − P0 (h)

600

400

400

200

200

10−2

10−1

100

101

102

103

104

105 t

P1 (h) − P1 (h)

800

600

0

13

0

10−2

P1 (h/2) − P1 (h)

10−1

100

101

102

103

104

105 t

Figure 5. Number of iteration required by th PCG scheme to solve the linear system ensuing form the elliptic equation, when µh ∈ P0,h (left panel) and µh ∈ P1h (right panel). On both panels the results obtained with and without the use of the sub-grid are included.

that, although far from optimal, our approach is potentially very effective and competitive with literature approaches in the solution of the Monge-Kantorovich equations. Before initiating this discussion, we note that our transient simulations show an initial phase (approximately until var(µkh ) ≈ 10−3 ) showing profiles of var(µkh ) and err(µkh ) that are superimposed for all mesh-refinements. This initial phase is characterized by strong variations of µh , during which the support Qµ starts to delineate with a fast decay of µh in the regions where the optimal density is zero. After this initial phase, µh varies more slowly and stabilizes within Qµ to its final value. At the same time, in Ω \ Qµ , the decay continues towards the final preset limit. This phase is characterized by larger time-step sizes and faster PCG convergence. We measure computational cost by looking at the total number of iterations needed by the PCG method to achieve convergence at each linear system solve, which is obtained when the relative residual is smaller than τCG . The linear solve phase is prevalent with respect to the matrix assembly phase, being approximately between 60 and 80% of the total cost. Moreover, it is the only phase whose cost varies with time, the assembly phase having a constant computational effort. We remark here that the nonlinearity of the problem and the time-variability of µh forces the reassembly of the stiffness matrix at each time-step for Explicit Euler, and at each Picard iteration for Implicit Euler. Figure 5 shows the number of PCG iterations to convergence vs. time in the solution of the linear system arising from the FEM discretization of the elliptic equation eq. (2a) using either the P1,h/p − P0,h (left panel) or P1,h/p − P1,h (right panel) approaches on the finest Mesh-1 refinement level (p = 1 or 2). Both methods based on a single mesh (p = 1) and on the use of the sub-mesh (p = 2) are reported in each panel. For both approaches the initial transient is the most expensive, with p = 2 characterized by a doubled number of linear iterations, reflecting the fact that the meshes are parametrized by the mesh parameter h used for the discretization of µh . After the initial transient the number of linear iterations starts decreasing, and tends to zero as time progresses, with a faster reduction for the two-mesh methods. Implicit Euler and convergence of the Picard scheme. In the case of implicit Euler time-stepping, the nonlinear system is solved by Picard iteration as described

14


S(µh (t))

(1)

µ0

(2)

µ0

2.0

(3)

µ0 1.0

10−1

100

101

t

Figure 6. Time behavior of the Lyapunov candidate function S(µkh ). in eq. (9) or in eq. (10). All numerical experiments show that the number of iterations of the Picard scheme increases linearly with the time step ∆tk , suggesting a fixed rate of contraction. We estimated it by computing the relative µh -variation defined as: ∗

(12)

C(k) := ∗

kµm h

,k

m∗ −1,k

kµh

∗

− µm h −

−1,k

kL2 (Ω) m∗ −2,k µh kL2 (Ω)

where m is the Picard iteration number when convergence occurs. Independently of the spatial discretization method used we obtain the following estimate C(k) ≈ K∆tk

K=1

This suggests that ∆tk can be used as a proxy to control the time-step evolution in the case of implicit Euler. Moreover we observe that with fixed time step ∆tk ,the number of iterations required by the Picard scheme monotonically decreases as we approach the equilibrium, and at the end of the time-iterations very few fix-point iterations are required by the Picard scheme to converge ( between 2 and 5 ). Dynamics of S (µ(t)). As a further verification to test the convergence towards steady state, we look at the time behavior of the Lyapunov-candidate functional S(µh (t)) for the different initial conditions described in eq. (11). Figure 6 reports this behavior for the finest mesh of set 2 of the P1,h/2 − P0,h method using a log scale in time. The results for other methods and mesh sets are practically indistinguishable, and are not reported here. We see that S decreases monotonically and always attains the same minimum value in time independently on the initial conditions. After t ≈ 100 the value of S(µh (t)) becomes approximately stationary, up to machine precision. 4.2. Test Case 2: comparison with literature solutions. Homogeneous case. In this section we propose a comparison with the results obtained by [1] using a finite element method to solve a regularized mixed formulation of the MK equations. Their discretization approach yields a highly nonlinear algebraic system that is solved using a modified successive over-relaxation method. They propose a set of test problems that we use here to practically show the characteristics of our method for the solution of the MK equations. We address the first test case proposed by [1], which considers the transport of a uniform density supported on a circle towards a disjoint ellipsis with the same area. Figure 7 (left) shows the domain Ω where problem eq. (2) is defined and the supports Q+ and


15

Q+ f+ Q− f−

Figure 7. Domain and supports of the forcing function used in the discretization of the MK equations eq. (2) for the solution of Example 1 of [1]. The two triangulations Th and Th/2 are also shown with blue and dashed black lines, respectively. Q− of the forcing term f = f + − f − , with f + (x) = 2 for x ∈ Q+ and zero otherwise, and f − (y), appropriately rescaled for y ∈ Q− to ensure mass balance. The coarse initial mesh is also shown in light blue lines, and its uniform refinement is shown in thin dashed lines. This mesh, characterized by 820 nodes and 1531 triangles, is a constrained Delaunay triangulation that follows the boundaries of both Q+ and Q− . The forcing function is adapted to this triangulation to enforce the compatibility condition of zero mean. The same Figure shows in the right panel the time-converged spatial distribution of the transport density numerically evaluated with the most stable discretization method, P1,h/2 − P0,h , on the finest mesh, which is used as a reference solution. The spatial distribution of µh is in good agreement with the results obtained by [1], achieving its maximum value (0.482) on the boundary of the left circle, and its minimum value 10−10 set by the prescribed lower bound as discussed in section 3.3. We would like to note that a similar test case was already proposed in [12] to verify the applicability and the robustness of this approach and to test the conjecture that the solution of the dynamic MK problem eq. (2) converges at infinite time towards the solution of the static MK equations. In this section we re-use this sample test to experimentally discuss the need to use different FEM spaces for the discretization of the transport density and of the transport potential. We start this discussion by presenting the results obtained using the P1,h − P0,h approach on the coarsest grid and look at three different times during the evolution. The times are selected so that var(µkh ) reaches the values 10−3 , 10−4 , 5 × 10−8 , namely t1 = 2.73 × 102 , t2 = 1.36 × 103 , and t3 = 2.5 × 105 , the latter time corresponding to the time-converged solution. We plot in fig. 8 both µh (upper panels) and | ∇ uh | (lower panels). At the first time the solution clearly resembles the reference solution shown in fig. 7 (right), although at a much coarser resolution. Correspondingly, the gradient in the second row displays some slight but acceptable overshoots in a region that resembles Qµ . Already at this early time, which occurs after 1630 time steps, some oscillations are visible. At time t2 these oscillations are much more pronounced with a checkerboard pattern that suggests an intrinsic instability of the scheme. We should note that the color scale in the plots are limited above and below by suitable values that emphasize the oscillations. The maximum and minimum values for both µh and | ∇ uh | are reported right below

16


Figure 8. Solution of Test Case 2 at t1 = 2.73 × 102 (left), t2 = 1.36 × 103 (center) , and t3 = 2.5 × 105 (right), using the P1,h − P0,h approach. Top row: spatial distribution of µh ∈ P0,h . Bottom row: spatial distribution of | ∇ uh | ∈ P0,h as calculated from uh ∈ P1,h .

Figure 9. Solution of Test Case 2 at t1 = 6.821 (left), t2 = 5.362 (center), and t3 = 9.65time103 (left), using the P1,h − P1,h approach. Top row: spatial distribution of µh ∈ P1,h . Bottom row: spatial distribution of | ∇ uh | ∈ P0,h calculated from uh ∈ P1,h . each legend. We observe that there are no overshoot in | ∇ uh |, which at the final time is never greater than one. Still, checkerboard-like oscillations are visible. These fluctuations cause the dynamic equation to drive µh to zero quickly, causing a drastic deterioration of the solution accuracy. The situation does not improve by using higher order spaces for µh . Figure 9 shows the results obtained by using a P1,h − P1,h approach. We still observe oscillations, albeit appearing at a later time and with a different pattern. Once oscillations in | ∇ uh | around the unit value start developing the dynamic equation determines


17

Figure 10. Solution of Test Case 2 at t1 = 6.75 × 101 (left), t2 = 2.04 × 102 (center), and t3 = 1.54 × 103 (right) using the P1,h/2 − P0,h approach. Top row: spatial distribution of µh ∈ P0,h . Middle row: spatial distribution of | ∇ uh | ∈ P0,h/2 calculated from uh ∈ P1,h/2 . Bottom row: spatial distribution of | ∇ uh | ∈ P0,h . a decay of µh within the elements where | ∇ uh | < 1 even if located within Qµ . This decay quickly reinforces in time leading to the observed checkerboard pattern. This behavior resembles the classical lack of stability due to a violation of an inf-sup-like constraint, but at this point we are not able to identify this condition. On the other hand, oscillations completely disappear if we employ a two-mesh approach. Looking at the checkerboard oscillations displayed in fig. 8, it is intuitive to think that averaging the gradient magnitude between neighboring triangles should compensate the fluctuations. This observation led us to employ the P1,h/2 − P0,h discretization described in section 3. Indeed, with this approach the gradients calculated from uh ∈ P1 (Th/2 ) are projected onto the space P0 (Th ) for insertion into the dynamic equation eq. (6b). This projection is equivalent to averaging the piecewise constant gradients over the four triangles of Th/2 that form one triangle of Th . This results in a oscillation free µh field, as shown in fig. 10. It is evident that no µh oscillations form even at the coarsest mesh level used in this test. Note that the spatial discretization of the elliptic equation does not guarantee monotonicity [15]. In fact, the gradient magnitudes arising from uh ∈ P1,h/2 still show the classical checkerboard fluctuations (fig. 10, middle row). However, the projection of | ∇ uh | onto P0,h (fig. 10, bottom row) does not show oscillations, albeit small overshooting occurs especially at the earlier times. We should emphasize that | ∇ uh | is plotted here using an extremely narrow color scale ranging within [0.9999, 1.0001]. Looking at the final time-converged solution, the value | ∇ u∗h | within the support of µ∗h and neighboring regions is exactly unitary, and remains bounded by 1 almost

18


everywhere, in compliance with the constraint of the MK equations. Only one small region with | ∇ u∗h | > 1 develop with a maximum value approaching 1.00009, considered consistent with the tolerance used in the PCG linear solve and the lower bound set on µh that cannot be smaller than 10−10 . Indeed, oscillations of the order of 10−5 in the gradient magnitude may be indistinguishable by the linear solver of the uh equation. Similar considerations can be done in the case µh ∈ P1,h (not shown here) but in this case some oscillations in | ∇ uh | persist even when uh ∈ P1,h/2 . This reinforces the conjecture that some sort of inf-sup stability condition exists that couples the discretization spaces for uh and µh , and will be the subject of further studies. One final observation for this test case concerns the computational cost of our approach. In comparison with the technique proposed by [1], our method seems to be computationally advantageous. In fact, as already mentioned, the simulations reported in [1] where obtained using a mixed FEM approach in combination with adaptive mesh refinement, leading to nonlinear systems of dimension approaching 60000. In our case, the dimensions for the smallest test case are 1531 (number of triangles in Th ) and 3170 (number of nodes in Th/2 ) for the diagonal dynamic algebraic system and the elliptic system, respectively, leading to a total of 4701 degrees of freedom. Note that the finest solution of fig. 7 was obtained with a total of 73917 degrees of freedom. Confidence on the proposed approach is reinforced by the observation that effective simulations can be obtained at intermediate mesh levels. Moreover, time-convergence can be considered achieved at much earlier times then the ones employed in this work if we look at the stationarity of the Lyapunov-candidate function. Obviously, adding simple adaptive mesh refinement strategies would greatly enhance the performance of the studied methodology.

5. Conclusions The performance of the proposed finite element method for the solution of the dynamic Monge-Kantorovich equations has been thoroughly analyzed experimentally on several test cases. The results show that the strategy of solving the MK equations by searching for the stationary solution of the dynamic MK problem is highly promising. The resulting system seems highly effective achieving experimentally optimal convergence in space and time even when using simple successive (Picard) linearization schemes. This is only a preliminary study, as the enhancement path for the proposed approach is clear. Some of the issues currently under study include the development of a Newton method for the solution of the nonlinear system in the case of implicit Euler time-stepping, to completely exploit the geometric convergence towards steady state only hinted at in the present paper. In this case, care must be exerted to avoid singularities of the Jacobian when µh → 0. Further improvements can be readily obtained by careful use of a sequence of meshes with progressively finer resolution as time increases, with adaptation to the support of the transport density easily achievable. Extension to the three-dimensional case is immediate, but is not presented in this work because of lack of reference solutions in this case. Theoretical work is needed to determine the exact relationships between the spatial discretization spaces used for uh and µh that guarantee stability of the approach. Future studies include the handling of less regular forcing functions to address the more interesting problems that can be studied by means of Optimal Transportation theory. We also believe that this work can be a useful starting point to further developments of time-dependent transport systems.


19

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